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Partial regularity for minimizers of variational integrals with discontinuous integrands. (English) Zbl 0863.35022

The author proves the partial regularity for vector-valued minimizers \(u\) of the variational integral \[ \int_\Omega [f(x,u,Du)+g(x,u)]dx, \] where \(f\) is strictly quasiconvex, of polynomial growth and continuous and \(g\) is a bounded Carathéodory function. Here \(\Omega\) is a bounded domain in \(\mathbb{R}^n\), \(n\geq 2\), \(u:\Omega\to\mathbb{R}^N\), \(N\geq 1\) and \(Du(x)\in \mathbb{R}^{N\times n}\) denotes the gradient of \(u\) at the point \(x\in\Omega\). An elementary proof for the special case of strict convexity and quadratic growth of \(f(x,u,\cdot)\) is presented.

MSC:

35B65 Smoothness and regularity of solutions to PDEs
49N60 Regularity of solutions in optimal control
35J50 Variational methods for elliptic systems
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References:

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